Subgroups — C₂×C₂₅₁⋊#338 #44550068

Order 32128.

372 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860794	C₂	1	2	yes	maximal
2	C₂₅₁⋊#338	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16860794	—	2	2	no	—
5	C₂×#16860794	—	2	2	no	—
6	C₂×#16860794	—	2	2	no	—
7	C₂×#16860794	—	2	2	no	—
8	C₂×#16860794	C₄	1	4	yes	—
9	C₂×#16860794	C₂²	1	4	yes	—
10	C₂²×C₂₅₁⋊C₈	C₄	1	4	yes	—
11	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₂₅₁	—	2	2	no	—
13	C₂×M₁₆×C₂₅₁	—	2	2	no	—
14	C₂×M₁₆×C₂₅₁	C₄	1	2	yes	—
15	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
16	C₂²×C₈×C₂₅₁	C₄	1	2	yes	—
17	#16860794	—	2	4	no	—
18	#16860794	—	2	4	no	—
19	#16860794	—	2	4	no	—
20	#16860794	—	2	4	no	—
21	C₂×C₂₅₁⋊C₈	—	2	4	no	—
22	C₂×C₂₅₁⋊C₈	—	2	4	no	—
23	C₂×C₂₅₁⋊C₈	C₂×C₄	1	8	yes	—
24	C₂⁴×C₂₅₁	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
30	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
31	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₅₁	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₅₁	—	2	4	no	—
36	M₁₆×C₂₅₁	—	2	4	no	—
37	C₂×C₈×C₂₅₁	—	2	2	no	—
38	C₂×C₈×C₂₅₁	—	2	2	no	—
39	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
40	C₂₅₁⋊C₈	—	2	8	no	—
41	C₂³×C₂₅₁	—	4	2	no	—
42	C₂³×C₂₅₁	C₄⋊C₄	1	2	yes	—
43	C₂³×C₂₅₁	C₄⋊C₄	1	2	yes	—
44	C₂³×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₅₁	C₄²	1	2	yes	—
48	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
49	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
50	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
65	C₈×C₂₅₁	—	2	4	no	—
66	C₂²×C₂₅₁	—	2	4	no	—
67	C₂²×C₂₅₁	—	2	2	no	—
68	C₂²×C₂₅₁	—	4	2	no	—
69	C₂²×C₂₅₁	—	4	1	no	—
70	C₂²×C₂₅₁	—	4	1	no	—
71	C₂²×C₂₅₁	#95	1	2	yes	—
72	C₂²×C₂₅₁	#95	1	1	yes	characteristic
73	C₂²×C₂₅₁	#95	1	1	yes	characteristic
74	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₂₅₁	#95	1	2	yes	—
77	C₄×C₂₅₁	#95	1	1	yes	characteristic
78	C₄×C₂₅₁	#95	1	1	yes	characteristic
79	C₄×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
80	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	2	yes	—
81	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₂₅₁	—	2	2	no	—
83	C₂×C₂₅₁	—	2	1	no	—
84	C₂×C₂₅₁	—	2	2	no	—
85	C₂×C₂₅₁	—	2	1	no	—
86	C₂×C₂₅₁	#338	1	2	yes	complemented
87	C₂×C₂₅₁	C₂×#95	1	1	yes	characteristic
88	C₂₅₁	C₂×#338	1	1	yes	characteristic, complemented, Sylow
89	C₂×#338	—	251	1	no	maximal, Sylow
90	C₂²×M₁₆	—	251	2	no	—
91	C₂²×M₁₆	D₂₅₁	1	1	yes	characteristic
92	#338	—	251	4	no	—
93	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	502	2	no	—
97	C₂×M₁₆	—	502	2	no	—
98	C₂×M₁₆	—	502	2	no	—
99	C₂×M₁₆	—	502	2	no	—
100	C₂×M₁₆	—	251	4	no	—
101	C₂×M₁₆	—	251	4	no	—
102	C₂×M₁₆	Dic₂₅₁	1	2	yes	—
103	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
104	C₂²×C₈	—	251	4	no	—
105	C₂²×C₈	Dic₂₅₁	1	2	yes	—
106	C₂⁴	Dic₅₀₂	1	1	yes	characteristic
107	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
108	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
109	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
112	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
113	C₂²×C₄	D₁₀₀₄	1	1	yes	characteristic
114	C₂²×C₄	D₁₀₀₄	1	1	yes	characteristic
115	C₂²×C₄	Dic₅₀₂	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
117	M₁₆	—	2	4	no	—
118	M₁₆	—	2	4	no	—
119	M₁₆	—	502	4	no	—
120	M₁₆	—	502	4	no	—
121	M₁₆	—	502	4	no	—
122	M₁₆	—	502	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	502	4	no	—
126	C₂×C₈	—	502	4	no	—
127	C₂×C₈	—	251	8	no	—
128	C₂×C₈	C₂×Dic₂₅₁	1	4	yes	—
129	C₂³	—	4	2	no	—
130	C₂³	Dic₂₅₁⋊C₄	1	2	yes	—
131	C₂³	Dic₂₅₁⋊C₄	1	2	yes	—
132	C₂³	C₄⋊Dic₂₅₁	1	1	yes	characteristic
133	C₂³	C₄⋊Dic₂₅₁	1	1	yes	characteristic
134	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
135	C₂×C₄	C₄×Dic₂₅₁	1	1	yes	characteristic
136	C₂×C₄	C₄×Dic₂₅₁	1	2	yes	—
137	C₂×C₄	C₄×Dic₂₅₁	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
140	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
141	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
142	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
144	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
145	C₂×C₄	Dic₂₅₁⋊C₄	1	2	yes	—
146	C₂×C₄	Dic₂₅₁⋊C₄	1	2	yes	—
147	C₂×C₄	C₄⋊Dic₂₅₁	1	1	yes	characteristic
148	C₂×C₄	C₄⋊Dic₂₅₁	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
150	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
152	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	—	502	8	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	C₂₅₁⋊#95	1	2	yes	—
161	C₂²	C₂₅₁⋊#95	1	1	yes	characteristic
162	C₂²	C₂₅₁⋊#95	1	1	yes	characteristic
163	C₂²	C₂×Dic₂₅₁⋊C₄	1	2	yes	—
164	C₂²	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
165	C₄	C₂₅₁⋊#95	1	2	yes	—
166	C₄	C₂₅₁⋊#95	1	1	yes	characteristic
167	C₄	C₂₅₁⋊#95	1	1	yes	characteristic
168	C₄	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
169	C₄	C₂×(C₂×D₂₅₁)⋊C₄	1	2	yes	—
170	C₄	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
171	C₂	—	2	2	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	2	no	—
174	C₂	—	2	1	no	—
175	C₂	C₂₅₁⋊#338	1	2	yes	direct product factor
176	C₂	C₂×C₂₅₁⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₂₅₁
commutator	C₄×C₂₅₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂²×M₁₆×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#95
prime radical	C₂⁴×C₂₅₁	Q₈
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂²×M₁₆	D₂₅₁
hypocenter	C₂₅₁	C₂×#338
LCS	G ▷ C₄×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂₅₁⋊#338	C₂	direct (N × H)
86. C₂×C₂₅₁	#338	semidirect
88. C₂₅₁	C₂×#338	semidirect
175. C₂	C₂₅₁⋊#338	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#16860794	2
C₂₅₁⋊#338	2
C₂²×M₁₆×C₂₅₁	2
C₂×#338	251

Subgroups — C2×C251⋊#338 #44550068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₅₁⋊#338 #44550068